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There are two times it is customary to read Shir HiShirim over Pesach: Once, by one's self, if he has the strength, after the seder (in Chutz LaAretz, twice) and the second time BiTzibur, as a congregation on Shabbos Chol HaMoed.

Shir Hashirim is a parable that describes a tremendous desire of a shepherd girl for her beloved King. She gets him, but loses him due to her own indiscretion. She earns him back only after intense searching and coming to a true recognition of her love for him not based on any ulterior motive. She gets Him back and reaches a point where the love is the same that it was at the start of their relationship. This parallels the experience of the Jewish people, starting with G-d's manifestation in the tents of our fore-mothers through the saga of Yoseph and his brothers, the enslavement in Egypt, the redemption, the receiving of the Torah, and the redemption finishing with the building of the Mishkan and G-d resting His presence in it.

On a more personal level, the actual relationship it is describing is the yearning of the Jewish soul for its Maker. After we have "experienced" our own personal liberation at the seder, we are not finished. We need to focus on building a place for G-d in our hearts, hence the custom to individually read Shir HaShirim after the seder [as a bridge for the next level of redemption that the end of Pesach deals with .... (my own thought)].
(Culled from Pachad Yitzchak, Pesach, Reshimos Vav and Zayin)

Maharal, in one of his prefaces to Gevuros Hashem, says that the Shabbos before any special time in the Jewish calendar is when the spiritual energy for that special time is given to this world. A m*ajor theme of the last day(s) of Pesach*, besides for the splitting of the Red Sea, is Ahdus, unity, and yearning for the final redemption.

We read Shir Hashirim the Shabbos before because Shir Hashirim, on a national level, talks about our relationship with G-d as the Jewish People. It tells the story of our struggle with accepting the idea that He actually has a more intimate relationship with us than other nations. It tells how, because of this disbelief in the specialness of the relationship, we messed up. It tells how we search for Him once He seems gone. It tells how we try to achieve redemption from all the wrong powers. (The Akeidas Yitzchak's words, not mine!) Ultimately that redemption will happen when we recognize that our entire relationship with Him is predicated on the fact that we and He love each other so much, as much as death! Only then will the ultimate redemption happen. (Culled from Akeidas Yitzchak on Shir HaShirim)

In summary, the first part of Pesach deals with a more personal redemption, hence the private reading of Shir Hashirim after the seder. It also is to help us connect this personal experience of redemption to a more macrocosmic understanding of redemption. The second part of Pesach deals with the ultimate redemption, hence on the Shabbos before (or in some years on the seventh day of Pesach) we read Shir Hashirim as a Tzibbur.